Problem 3 - Thomas Calculus

Question

Write the sums without sigma notation. Then evaluate them. $$ \sum_{k=1}^{4} \cos k \pi $$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Understanding the Sigma Notation

The sum given is \( \sum_{k=1}^{4} \cos k \pi \), which means we need to evaluate the cosine of \( k \pi \) for \( k = 1, 2, 3, \) and \( 4 \), then sum these values.

Step 2: Expanding the Sigma Notation

Write out the terms without using sigma notation: \( \cos(1 \pi) + \cos(2 \pi) + \cos(3 \pi) + \cos(4 \pi) \).

Step 3: Evaluate Each Term

Calculate the cosine for each term separately:- \( \cos(1 \pi) = \cos(\pi) = -1 \)- \( \cos(2 \pi) = 1 \)- \( \cos(3 \pi) = \cos(\pi) = -1 \)- \( \cos(4 \pi) = 1 \)

Step 4: Calculate the Sum

Add up all the evaluated terms: \( -1 + 1 + (-1) + 1 = 0 \).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are essential in various fields, including physics, engineering, and astronomy. There are six primary trigonometric functions, with the three most common being sine, cosine, and tangent.

• The **sine function** (\( ext{sin}\)) takes an angle and returns the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• The **cosine function** (\( ext{cos}\)) returns the ratio of the adjacent side to the hypotenuse.
• The **tangent function** (\( ext{tan}\)) gives the ratio of the opposite side to the adjacent side.

Trigonometric functions are periodic, meaning they repeat their values in regular intervals, making them useful for modeling periodic phenomena such as sound and light waves.
When dealing with angles, particularly those expressed in radians, understanding these functions becomes crucial, especially in terms of evaluating expressions that involve them.

Exploring the Cosine Function

The cosine function is a fundamental part of trigonometry, defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. It can be expressed for any angle \(\theta\) as \(\cos \theta\).
The cosine function:

• is an even function, which means \(\cos(-\theta) = \cos(\theta)\), making it symmetric about the vertical axis.
• has a period of \(2\pi\), implying that its graph repeats every \(2\pi\) units.
• oscillates between -1 and 1, with \(\cos(0) = 1\) and points where \(\cos(\pi) = -1\).

When calculating expressions such as the sum \(\cos k\pi\) for different values of \(k\), it's important to note how the cosine values alternate:

• For \(k = 1, 3, 5, \ldots\) (\(k\) is odd), \(\cos(k\pi) = -1\).
• For \(k = 2, 4, 6, \ldots\) (\(k\) is even), \(\cos(k\pi) = 1\).

This pattern helps simplify calculations when evaluating sums involving multiple cosine terms.

Sum Evaluation in Trigonometric Contexts

The process of evaluating sums in trigonometry often involves expanding expressions and calculating the value of each trigonometric function involved. With sigma notation, this becomes systematic, where the sum signifies a series of terms generated from a common pattern.
When we expand a sum like \(\sum_{k=1}^{4} \cos k \pi\), we rewrite it without sigma notation as \(\cos(1\pi) + \cos(2\pi) + \cos(3\pi) + \cos(4\pi)\).
This step-by-step expansion allows you to evaluate each term individually:

• \(\cos(1 \pi) = -1\)
• \(\cos(2 \pi) = 1\)
• \(\cos(3 \pi) = -1\)
• \(\cos(4 \pi) = 1\)

After evaluating each cosine term, add them to find the sum: \(-1 + 1 - 1 + 1 = 0\).
This computation highlights the usefulness of recognizing patterns within trigonometric functions to simplify otherwise complex sum evaluations.